Properties

Label 10.18.a.c
Level 10
Weight 18
Character orbit 10.a
Self dual yes
Analytic conductor 18.322
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3222087345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{83281}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 30\sqrt{83281}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -256 q^{2} + ( -654 - \beta ) q^{3} + 65536 q^{4} + 390625 q^{5} + ( 167424 + 256 \beta ) q^{6} + ( 301922 - 2907 \beta ) q^{7} -16777216 q^{8} + ( -53759547 + 1308 \beta ) q^{9} +O(q^{10})\) \( q -256 q^{2} + ( -654 - \beta ) q^{3} + 65536 q^{4} + 390625 q^{5} + ( 167424 + 256 \beta ) q^{6} + ( 301922 - 2907 \beta ) q^{7} -16777216 q^{8} + ( -53759547 + 1308 \beta ) q^{9} -100000000 q^{10} + ( -235740648 + 5238 \beta ) q^{11} + ( -42860544 - 65536 \beta ) q^{12} + ( -770917114 + 87228 \beta ) q^{13} + ( -77292032 + 744192 \beta ) q^{14} + ( -255468750 - 390625 \beta ) q^{15} + 4294967296 q^{16} + ( 16069950282 - 4460868 \beta ) q^{17} + ( 13762444032 - 334848 \beta ) q^{18} + ( 64336264700 + 7351668 \beta ) q^{19} + 25600000000 q^{20} + ( 217690623312 + 1599256 \beta ) q^{21} + ( 60349605888 - 1340928 \beta ) q^{22} + ( 325179929646 - 7606701 \beta ) q^{23} + ( 10972299264 + 16777216 \beta ) q^{24} + 152587890625 q^{25} + ( 197354781184 - 22330368 \beta ) q^{26} + ( 21578017140 + 182044278 \beta ) q^{27} + ( 19786760192 - 190513152 \beta ) q^{28} + ( 1271527374990 - 444821544 \beta ) q^{29} + ( 65400000000 + 100000000 \beta ) q^{30} + ( 3919703999372 + 461196486 \beta ) q^{31} -1099511627776 q^{32} + ( -238428906408 + 232314996 \beta ) q^{33} + ( -4113907272192 + 1141982208 \beta ) q^{34} + ( 117938281250 - 1135546875 \beta ) q^{35} + ( -3523185672192 + 85721088 \beta ) q^{36} + ( -13902604548778 - 818505288 \beta ) q^{37} + ( -16470083763200 - 1882027008 \beta ) q^{38} + ( -6033811768644 + 713870002 \beta ) q^{39} -6553600000000 q^{40} + ( -18913182132078 + 4473032724 \beta ) q^{41} + ( -55728799567872 - 409409536 \beta ) q^{42} + ( 14815226963426 - 7033515453 \beta ) q^{43} + ( -15449499107328 + 343277568 \beta ) q^{44} + ( -20999823046875 + 510937500 \beta ) q^{45} + ( -83246061989376 + 1947315456 \beta ) q^{46} + ( 110395791511002 + 18068763213 \beta ) q^{47} + ( -2808908611584 - 4294967296 \beta ) q^{48} + ( 400861292338977 - 1755374508 \beta ) q^{49} -39062500000000 q^{50} + ( 323845245632772 - 13152542610 \beta ) q^{51} + ( -50522823983104 + 5716574208 \beta ) q^{52} + ( 529153008800286 - 14912836812 \beta ) q^{53} + ( -5523972387840 - 46603335168 \beta ) q^{54} + ( -92086190625000 + 2046093750 \beta ) q^{55} + ( -5065410609152 + 48771366912 \beta ) q^{56} + ( -593104753551000 - 69144255572 \beta ) q^{57} + ( -325511007997440 + 113874315264 \beta ) q^{58} + ( 763910358617580 + 122690464632 \beta ) q^{59} + ( -16742400000000 - 25600000000 \beta ) q^{60} + ( -315869106561058 - 134737202448 \beta ) q^{61} + ( -1003444223839232 - 118066300416 \beta ) q^{62} + ( -301228798981734 + 156673917105 \beta ) q^{63} + 281474976710656 q^{64} + ( -301139497656250 + 34073437500 \beta ) q^{65} + ( 61037800040448 - 59472638976 \beta ) q^{66} + ( -4421881746936298 + 197259831009 \beta ) q^{67} + ( 1053160261681152 - 292347445248 \beta ) q^{68} + ( 357476625394416 - 320205147192 \beta ) q^{69} + ( -30192200000000 + 290700000000 \beta ) q^{70} + ( -1665950830456188 - 254971321938 \beta ) q^{71} + ( 901935532081152 - 21944598528 \beta ) q^{72} + ( -5013175287146014 + 248953009068 \beta ) q^{73} + ( 3559066764487168 + 209537353728 \beta ) q^{74} + ( -99792480468750 - 152587890625 \beta ) q^{75} + ( 4216341443379200 + 481798914048 \beta ) q^{76} + ( -1212473052536856 + 686879531172 \beta ) q^{77} + ( 1544655812772864 - 182750720512 \beta ) q^{78} + ( 4131803421027560 - 152690394132 \beta ) q^{79} + 1677721600000000 q^{80} + ( -6716341925329599 - 309550308156 \beta ) q^{81} + ( 4841774625811968 - 1145096377344 \beta ) q^{82} + ( -5344763446168494 - 196076217465 \beta ) q^{83} + ( 14266572689375232 + 104808841216 \beta ) q^{84} + ( 6277324328906250 - 1742526562500 \beta ) q^{85} + ( -3792698102637056 + 1800579955968 \beta ) q^{86} + ( 32509085802034140 - 980614085214 \beta ) q^{87} + ( 3955071771475968 - 87879057408 \beta ) q^{88} + ( 27939206260952490 + 3148403792040 \beta ) q^{89} + ( 5375954700000000 - 130800000000 \beta ) q^{90} + ( -19238698305301508 + 2267392102614 \beta ) q^{91} + ( 21310991869280256 - 498512756736 \beta ) q^{92} + ( -37131500511098688 - 4221326501216 \beta ) q^{93} + ( -28261322626816512 - 4625603382528 \beta ) q^{94} + ( 25131353398437500 + 2871745312500 \beta ) q^{95} + ( 719080604565504 + 1099511627776 \beta ) q^{96} + ( -28053290181929878 + 10663485822180 \beta ) q^{97} + ( -102620490838778112 + 449375874048 \beta ) q^{98} + ( 13186835549548056 - 589941274770 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 512q^{2} - 1308q^{3} + 131072q^{4} + 781250q^{5} + 334848q^{6} + 603844q^{7} - 33554432q^{8} - 107519094q^{9} + O(q^{10}) \) \( 2q - 512q^{2} - 1308q^{3} + 131072q^{4} + 781250q^{5} + 334848q^{6} + 603844q^{7} - 33554432q^{8} - 107519094q^{9} - 200000000q^{10} - 471481296q^{11} - 85721088q^{12} - 1541834228q^{13} - 154584064q^{14} - 510937500q^{15} + 8589934592q^{16} + 32139900564q^{17} + 27524888064q^{18} + 128672529400q^{19} + 51200000000q^{20} + 435381246624q^{21} + 120699211776q^{22} + 650359859292q^{23} + 21944598528q^{24} + 305175781250q^{25} + 394709562368q^{26} + 43156034280q^{27} + 39573520384q^{28} + 2543054749980q^{29} + 130800000000q^{30} + 7839407998744q^{31} - 2199023255552q^{32} - 476857812816q^{33} - 8227814544384q^{34} + 235876562500q^{35} - 7046371344384q^{36} - 27805209097556q^{37} - 32940167526400q^{38} - 12067623537288q^{39} - 13107200000000q^{40} - 37826364264156q^{41} - 111457599135744q^{42} + 29630453926852q^{43} - 30898998214656q^{44} - 41999646093750q^{45} - 166492123978752q^{46} + 220791583022004q^{47} - 5617817223168q^{48} + 801722584677954q^{49} - 78125000000000q^{50} + 647690491265544q^{51} - 101045647966208q^{52} + 1058306017600572q^{53} - 11047944775680q^{54} - 184172381250000q^{55} - 10130821218304q^{56} - 1186209507102000q^{57} - 651022015994880q^{58} + 1527820717235160q^{59} - 33484800000000q^{60} - 631738213122116q^{61} - 2006888447678464q^{62} - 602457597963468q^{63} + 562949953421312q^{64} - 602278995312500q^{65} + 122075600080896q^{66} - 8843763493872596q^{67} + 2106320523362304q^{68} + 714953250788832q^{69} - 60384400000000q^{70} - 3331901660912376q^{71} + 1803871064162304q^{72} - 10026350574292028q^{73} + 7118133528974336q^{74} - 199584960937500q^{75} + 8432682886758400q^{76} - 2424946105073712q^{77} + 3089311625545728q^{78} + 8263606842055120q^{79} + 3355443200000000q^{80} - 13432683850659198q^{81} + 9683549251623936q^{82} - 10689526892336988q^{83} + 28533145378750464q^{84} + 12554648657812500q^{85} - 7585396205274112q^{86} + 65018171604068280q^{87} + 7910143542951936q^{88} + 55878412521904980q^{89} + 10751909400000000q^{90} - 38477396610603016q^{91} + 42621983738560512q^{92} - 74263001022197376q^{93} - 56522645253633024q^{94} + 50262706796875000q^{95} + 1438161209131008q^{96} - 56106580363859756q^{97} - 205240981677556224q^{98} + 26373671099096112q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
144.792
−143.792
−256.000 −9311.53 65536.0 390625. 2.38375e6 −2.48655e7 −1.67772e7 −4.24355e7 −1.00000e8
1.2 −256.000 8003.53 65536.0 390625. −2.04890e6 2.54694e7 −1.67772e7 −6.50836e7 −1.00000e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.18.a.c 2
3.b odd 2 1 90.18.a.k 2
4.b odd 2 1 80.18.a.d 2
5.b even 2 1 50.18.a.f 2
5.c odd 4 2 50.18.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.c 2 1.a even 1 1 trivial
50.18.a.f 2 5.b even 2 1
50.18.b.d 4 5.c odd 4 2
80.18.a.d 2 4.b odd 2 1
90.18.a.k 2 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1308 T_{3} - 74525184 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 256 T )^{2} \)
$3$ \( 1 + 1308 T + 183755142 T^{2} + 168915333204 T^{3} + 16677181699666569 T^{4} \)
$5$ \( ( 1 - 390625 T )^{2} \)
$7$ \( 1 - 603844 T - 168048464563602 T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + 471481296 T + 1064411254083979846 T^{2} + \)\(23\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 + 1541834228 T + 17324849107520411262 T^{2} + \)\(13\!\cdots\!24\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 - 32139900564 T + \)\(42\!\cdots\!78\)\( T^{2} - \)\(26\!\cdots\!28\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - 128672529400 T + \)\(11\!\cdots\!78\)\( T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - 650359859292 T + \)\(38\!\cdots\!22\)\( T^{2} - \)\(91\!\cdots\!76\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 - 2543054749980 T + \)\(13\!\cdots\!18\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - 7839407998744 T + \)\(44\!\cdots\!06\)\( T^{2} - \)\(17\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + 27805209097556 T + \)\(10\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!52\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + 37826364264156 T + \)\(40\!\cdots\!46\)\( T^{2} + \)\(98\!\cdots\!36\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - 29630453926852 T + \)\(82\!\cdots\!62\)\( T^{2} - \)\(17\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 220791583022004 T + \)\(41\!\cdots\!78\)\( T^{2} - \)\(58\!\cdots\!48\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - 1058306017600572 T + \)\(67\!\cdots\!22\)\( T^{2} - \)\(21\!\cdots\!36\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 1527820717235160 T + \)\(19\!\cdots\!38\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 + 631738213122116 T + \)\(32\!\cdots\!06\)\( T^{2} + \)\(14\!\cdots\!36\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + 8843763493872596 T + \)\(38\!\cdots\!58\)\( T^{2} + \)\(97\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + 3331901660912376 T + \)\(57\!\cdots\!26\)\( T^{2} + \)\(98\!\cdots\!16\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + 10026350574292028 T + \)\(11\!\cdots\!02\)\( T^{2} + \)\(47\!\cdots\!84\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - 8263606842055120 T + \)\(37\!\cdots\!18\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 + 10689526892336988 T + \)\(86\!\cdots\!82\)\( T^{2} + \)\(45\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 - 55878412521904980 T + \)\(27\!\cdots\!58\)\( T^{2} - \)\(77\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 + 56106580363859756 T + \)\(41\!\cdots\!58\)\( T^{2} + \)\(33\!\cdots\!72\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \)
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